# Efficient methods for Gaussian Markov random fields under sparse linear constraints

arXiv: https://arxiv.org/abs/2106.01712 · HTML: https://mlsanity.com/paper/2106.01712/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 43 references that are in the map.
- First posted on arXiv 3 Jun 2021.

## Builds on (most cited first, 10 of 10)

- [Randomized Block Krylov Methods for Stronger and Faster Approximate Singular Value Decomposition](https://mlsanity.com/paper/1504.05477/) (2015)
- [Linearly constrained Gaussian processes](https://mlsanity.com/paper/1703.00787/) (2017)
- [Modeling and interpolation of the ambient magnetic field by Gaussian processes](https://mlsanity.com/paper/1509.04634/) (2015)
- [LazySVD: Even Faster SVD Decomposition Yet Without Agonizing Pain](https://mlsanity.com/paper/1607.03463/) (2016)
- [Algorithmic Linearly Constrained Gaussian Processes](https://mlsanity.com/paper/1801.09197/) (2018)
- [Linearly Constrained Gaussian Processes with Boundary Conditions](https://mlsanity.com/paper/2002.00818/) (2020)
- [Deep Gaussian Markov Random Fields](https://mlsanity.com/paper/2002.07467/) (2020)
- [The rational SPDE approach for Gaussian random fields with general smoothness](https://mlsanity.com/paper/1711.04333/) (2017)
- [Spatial modelling with R-INLA: A review](https://mlsanity.com/paper/1802.06350/) (2018)
- [Fast Bayesian whole-brain fMRI analysis with spatial 3D priors](https://mlsanity.com/paper/1606.00980/) (2016)

## Built on by (most cited first, 1 of 1)

- [Fast approximate estimation of conditional Shapley values when using a linear explainer](https://mlsanity.com/paper/2504.18167/) (2025)

## Abstract (arXiv)

Methods for inference and simulation of linearly constrained Gaussian Markov Random Fields (GMRF) are computationally prohibitive when the number of constraints is large. In some cases, such as for intrinsic GMRFs, they may even be unfeasible. We propose a new class of methods to overcome these challenges in the common case of sparse constraints, where one has a large number of constraints and each only involves a few elements. Our methods rely on a basis transformation into blocks of constrained versus non-constrained subspaces, and we show that the methods greatly outperform existing alternatives in terms of computational cost. By combining the proposed methods with the stochastic partial differential equation approach for Gaussian random fields, we also show how to formulate Gaussian process regression with linear constraints in a GMRF setting to reduce computational cost. This is illustrated in two applications with simulated data.